Physics
Particles in Magnetic Fields
When charged particles move through a magnetic field, they experience a force perpendicular to both their velocity and the magnetic field direction. This force causes the particles to move in circular or helical paths, depending on their initial velocity. The behavior of particles in magnetic fields is fundamental to understanding phenomena such as the motion of charged particles in particle accelerators and the formation of auroras.
Written by Perlego with AI-assistance
Related key terms
1 of 5
11 Key excerpts on "Particles in Magnetic Fields"
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
One of the important and exciting areas in physics today is the study of elementary particles, which are the basic building blocks from which all matter is constructed. Important information about an elementary particle can be obtained from its motion in a magnetic field, with the aid of a device known as a bubble chamber. A bubble chamber contains a superheated liquid such as hydrogen, which will boil and form bubbles readily. When an electrically charged particle passes through the chamber, a thin track of bubbles is left in its wake. This track can be photographed to show how a magnetic field affects the particle motion. Conceptual Example 4 illustrates how physicists deduce information from such photographs. 598 CHAPTER 21 Magnetic Forces and Magnetic Fields CONCEPTUAL EXAMPLE 4 Particle Tracks in a Bubble Chamber Figure 21.13a shows the bubble-chamber tracks resulting from an event that begins at point A. At this point a gamma ray (emitted by certain radioactive substances), traveling in from the left, spontaneously trans- forms into two charged particles. There is no track from the gamma ray itself. These particles move away from point A, producing the two spiral tracks. A third charged particle is knocked out of a hydrogen atom and moves forward, producing the long track with the slight upward curvature. Each of the three particles has the same mass and carries a charge of the same magnitude. A uniform magnetic field is directed out of the paper to- ward you. What is the sign (+ or −) of the charge carried by each particle? Particle 1 Particle 2 Particle 3 (a) − − + (b) − + − (c) + − − (d) + − + Reasoning Figure 21.13b shows a positively charged particle traveling with a velocity v → that is perpendicular to a magnetic field. The field is directed out of the paper, just like it is in part a of the drawing. RHR-1 indicates that the magnetic force points downward.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
Important information about an elementary particle can be obtained from its motion in a magnetic field, with the aid of a device known as a bubble chamber. A bubble chamber contains a superheated liquid such as hydrogen, which will boil and form bubbles readily. When an electrically charged particle passes through the chamber, a thin track of bubbles is left in its wake. This track can be photographed to show how a magnetic field affects the particle motion. 584 Physics Conceptual example 4 illustrates how physicists deduce information from such photographs. CONCEPTUAL EXAMPLE 4 Particle tracks in a bubble chamber Figure 21.13a shows the bubble‐chamber tracks resulting from an event that begins at point A. At this point a gamma ray (emitted by certain radioactive substances), travelling in from the left, spontaneously transforms into two charged particles. There is no track from the gamma ray itself. These particles move away from point A, producing the two spiral tracks. A third charged particle is knocked out of a hydrogen atom and moves forwards, producing the long track with the slight upward curvature. Each of the three particles has the same mass and carries a charge of the same magnitude. A uniform magnetic field is directed out of the paper towards you. What is the sign (+ or −) of the charge carried by each particle? Particle 1 Particle 2 Particle 3 (a) − − + (b) − + − (c) + − − (d) + − + Reasoning Figure 21.13b shows a positively charged particle travelling with a velocity v that is perpendicular to a magnetic field. The field is directed out of the paper, just like it is in part a of the drawing. RHR‐1 indicates that the magnetic force points downwards. This magnetic force provides the centripetal force that causes a particle to move on a circular path (see section 5.3). The centripetal force is directed towards the centre of the circular path.
eBook - PDFQuantum Mechanics
A Modern Development
- Leslie E Ballentine(Author)
- 1998(Publication Date)
- WSPC(Publisher)
Chapter 11 Charged Particle in a Magnetic Field The theory of the motion of a charged particle in a magnetic field presents several difficult and unintuitive features. The derivation of the quantum theory does not require the classical theory; nevertheless it is useful to first review the classical theory in order to show that some of these unintuitive features are not peculiar to the quantum theory, but rather that they are characteristic of motion in a magnetic field. 11.1 Classical Theory The electric and magnetic fields, E and B, enter the Lagrangian and Hamil-tonian forms of mechanics through the vector and scalar potentials, A and : E= ~ V0 ~^fr (llla) B = V x A . (11.1b) (The speed of light c appears only because of a conventional choice of units.) The potentials are not unique. The fields E arid B are unaffected by the replacement 1 8Y A ^ A ' = A + V X , ^ 4! = **--£> ( 1L2 ) where — x( x > 0 i s a n arbitrary scalar function. This change of the potentials, called a gauge transformation, has no effect upon any physical result. It thus appears, in classical mechanics, that the potentials are only a mathematical construct having no direct physical significance. The Lagrangian for a particle of mass M and charge q in an arbitrary electromagnetic fields is £(x, v, t) = ^ -q (x, t) + ± v A (x, t), (11.3) tL C 307 308 Ch. 11: Charged Particle in a Magnetic Field where x and v = dx./dt are the position and velocity of the particle.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
524 Chapter 21 | Magnetic Forces and Magnetic Fields We have seen that a charged particle traveling in a magnetic field experiences a mag- netic force that is always perpendicular to the field. In contrast, the force applied by an elec- tric field is always parallel (or antiparallel) to the field direction. Because of the difference in the way that electric and magnetic fields exert forces, the work done on a charged particle by each field is different, as we now discuss. The Work Done on a Charged Particle Moving Through Electric and Magnetic Fields In Figure 21.9a an electric field applies a force to a positively charged particle, and, con- sequently, the path of the particle bends in the direction of the force. Because there is a component of the particle’s displacement in the direction of the electric force, the force does work on the particle, according to Equation 6.1. This work increases the kinetic en- ergy and, hence, the speed of the particle, in accord with the work–energy theorem (see Section 6.2). In contrast, the magnetic force in Figure 21.9b always acts in a direction that is perpendicular to the motion of the charge. Consequently, the displacement of the moving charge never has a component in the direction of the magnetic force. As a result, the magnetic force cannot do work and change the kinetic energy of the charged particle in Figure 21.9b. Thus, the speed of the particle does not change, although the force does alter the direction of the motion. The Circular Trajectory To describe the motion of a charged particle in a constant magnetic field more completely, let’s discuss the special case in which the velocity of the particle is perpendicular to a uniform magnetic field. As Figure 21.11 illustrates, the magnetic force serves to move the particle in a circular path. To understand why, consider two points on the circumference labeled 1 and 2.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
21.3 | The Motion of a Charged Particle in a Magnetic Field 585 21.3 | The Motion of a Charged Particle in a Magnetic Field Comparing Particle Motion in Electric and Magnetic Fields The motion of a charged particle in an electric field is noticeably different from the motion in a magnetic field. For example, Figure 21.9a shows a positive charge moving between the plates of a parallel plate capacitor. Initially, the charge is moving perpendicular to the direction of the electric field. Since the direction of the electric force on a positive charge is in the same direction as the electric field, the particle is deflected sideways. Part b of the drawing shows the same particle traveling initially at right angles to a magnetic field. An application of RHR-1 shows that when the charge enters the field, the charge is deflected upward (not sideways) by the magnetic force. As the charge moves upward, the direction of the magnetic force changes, always remaining perpendicular to both the magnetic field and the velocity. Conceptual Example 2 focuses on the difference in how electric and magnetic fields apply forces to a moving charge. CONCEPTUAL EXAMPLE 2 | The Physics of a Velocity Selector A velocity selector is a device for measuring the velocity of a charged particle. The device operates by applying electric and magnetic forces to the particle in such a way that these forces balance. Figure 21.10a shows a particle with a positive charge 1q and a velocity v B that is per- pendicular to a constant magnetic field* B B . Figure 21.10b illustrates a velocity selector, which is a cylindrical tube that is located within the magnetic field. Inside the tube there is a parallel plate capacitor that produces an electric field E B (not shown) perpendicular to the magnetic field. The charged particle enters the left end of the tube, moving perpendicular to the magnetic field.
eBook - PDFAdhesive Particle Flow
A Discrete-Element Approach
- Jeffery S. Marshall, Shuiqing Li(Authors)
- 2014(Publication Date)
- Cambridge University Press(Publisher)
The potential field and permit- tivity are denoted, respectively, by I and ε I interior to the body and by E and ε E exterior to the body. interest either behave in a nonlinear manner, where the volume magnetization vector M is not proportional to H, or else exhibit eddy currents as a response to fluctuating magnetic fields, and this analogy breaks down. In general, the problem of particles in electric and magnetic fields exhibits a rich physics, particularly when consideration is given to nonspherical or nonhomogeneous particles, but such topics are outside the scope of the current book. More extensive consideration of the electromechanics of particles of different types exposed to electric and magnetic fields can be found in Jones (1995). 8.4. Boundary Element Method Section 8.1 discusses how particles, even if they carry no charge, are sensitive to the electric field vector and its gradient. In conducting a DEM simulation with charged particles, it is therefore necessary to compute the electric field to which the particles are exposed. This electric field is influenced by two features of the flow field – the domain boundaries and the presence of other particles. Domain boundaries may include electrode surfaces that emit an electric field into the flow, or dielectric surfaces that nevertheless alter the electric field within the domain. An electric field may also be introduced by the presence of charged particles, each of which generates an electric field that decays slowly with distance away from the particle. Even if the other particles are uncharged, they will exhibit an induced dipole in the presence of an electric field that will influence forces and torques on nearby particles.
eBook - PDF- George Schmidt(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
Motion of Charged Particles in Electromagnetic Fields 2-1. The Static Magnetic Field For one particle moving in a magnetic field with velocity v, (1-7) reduces to As the force is perpendicular to the velocity, no work is done by the magnetic field. Indeed, the scalar multiplication of (2-1) by v yields showing that the kinetic energy of the particle, in an arbitrary magnetic field, is a constant of motion. Let us restrict ourselves for the moment to the special case where the magnetic field lines are straight and parallel (but the field is not necessarily uniform). Denoting vector components parallel to the field with the sub-script || and those perpendicular to it with _L, we obtain m — = q (vxB ) (2-1) ( 2 -2 ) V=V||+V1 (2-3) and (2-1) becomes (2-4) since v(|x Bvanishes. Equation(2-4) splits into a ||-component equation and a _L-component equation dy and dv II dy, q . -J + — ^= -(vj. x B) dt dt m = 0 (v || = const) £ -2 · < ν , χ Β ) at m 5 (2-5) (2-6) 6 II. Motion of Charged Particles Since the right side of (2-6) is perpendicular to v± , the left side is a centri-petal acceleration. It can be written ^ ( -r ) = ^ ( v ± xB) (2-7) where r is the local radius of curvature of the particle path (Fig. 2-1). Its F ig . 2-1. Particle moving in a magnetic field of straight and parallel field lines. The field intensity varies in the plane perpendicular to B. value is, from (2-7), r = mvL qB ( 2 -8 ) In the special case of a uniform magnetic field, B = const, and considering the constancy of v± from (2-2) and (2-5), the radius of curvature R = mv I qB (2-9) is also a constant. In a uniform magnetic field, therefore, the particle moves in a circle with the so-called cyclotron or gyroradius R in the perpendicular plane, while it moves with a constant velocity along the field lines.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
653 Active regions on the surface of the sun correspond to areas of intense magnetic fields. The looping magnetic field lines above this area are illuminated by the motion of charged particles. The moving charges experience a force in the magnetic field and spiral along them. In this chapter, we will study magnetic fields and the forces they apply to charged particles. The sizes of these magnetic structures can be enormous, as the earth, drawn to scale at the lower right, demonstrates. LEARNING OBJECTIVES After reading this module, you should be able to... 21.1 Define magnetic field. 21.2 Calculate the magnetic force on a moving charge in a magnetic field. 21.3 Analyze the motion of a charged particle in a magnetic field. 21.4 Describe how the masses of ions are determined using a mass spectrometer. 21.5 Calculate the magnetic force on a current in a magnetic field. 21.6 Calculate the torque on a current-carrying coil. 21.7 Calculate magnetic fields produced by currents. 21.8 Apply Ampère’s law to calculate the magnetic field due to a steady current. 21.9 Describe magnetic materials. Magnetic Forces and Magnetic Fields CHAPTER 21 21.1 Magnetic Fields Permanent magnets have long been used in navigational compasses. As Figure 21.1 illustrates, the compass needle is a permanent magnet sup- ported so it can rotate freely in a plane. When the compass is placed on a horizontal surface, the needle rotates until one end points approximately to the north. The end of the needle that points north is labeled the north magnetic pole; the opposite end is the south magnetic pole. Magnets can exert forces on each other. Figure 21.2 shows that the magnetic forces between north and south poles have the property that like poles repel each other, and unlike poles attract. This behavior is similar to that of like and unlike electric charges. How- ever, there is a significant difference between magnetic poles and electric charges.
eBook - PDFAtmosphere, Ocean and Climate Dynamics
An Introductory Text
- John Marshall, R. Alan Plumb(Authors)
- 1961(Publication Date)
- Academic Press(Publisher)
74 3. FIELDS, PARTICLES AND THE ATMOSPHERE where v is the velocity, m the mass, and e the absolute value of the charge on the parti~le.~ The plus sign applies to positive ions, the minus sign to electrons. The vector [e/c) v x B is the Lorentz force. If there is no electric field, the energy of the particle remains constant. To show this we take the scalar product of the velocity times Eq. (3.16): (3.17) where4 v = I v 1 . Thus, as the acceleration is always perpendicular to the velocity, the scalar speed does not change. 3.2.1. Uniform Magnetic Field In a uniform magnetic field a particle orbit consists of uniform motion along the field lines plus a circular motion in a plane perpendicular to the field. In a right-handed system of coordinates (see Fig. 3.1) with the magnetic field along the z axis, a positive particle gyrates in the - 4 direction, a negative particle in the opposite direction. That is, a particle tends to circle an external field in the direction such that the small magnetic field produced by the particle is in the direction opposite to the external field. Equating centrifugal force to the Lorentz force, we find for the angular velocity, (3.18) Here vL is the absolute value of the velocity component perpendicular to the field, wc is the gyrofrequency or cyclotron frequency, and p is the radius of gyration. For a proton in the Earth’s field near the surface (about 0.5 gauss), wC m 5 x lo3 radian/sec. If vl = lo9 cm/sec, the radius is p e 2 km. For an electron with a comparable velocity the orbital radius is much smaller. The product Bp (= mv,c/e) is often designated the magnetic rigidity. The symbol B (strictly, the magnetic induction) is used here rather than H (magnetic field strength). Because we are concerned here with the interactions of charged particles and magnetic fields, the induction is the physical quantity we are usually interested in. However, when we deal with currents we shall agree to consider the magnetization current J’ as incorporated in the (total) current J. Ordinarily V x B = 4a(J + J’) and V x H = 4nJ; but with our convention of
eBook - PDF- Hermann Wollnik(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
T h e f o r c e s F g g a n d a r e b o t h i n v o l t a m p e r e s e c o n d s p e r m e t e r . 30 2 Motion of Charged Particles in Electromagnetic Fields Fig. 2.1. Spatial relation between the velocity ν of a positively charged particle, the direction of the magnetic flux density B, and the Lorentz force F^. In order to find the direction of ¥j ß in space, it is a trivial but useful rule to form a tripod with the thumb, the forefinger, and the middle finger of the right hand. If the thumb points in the direction of ν and the forefinger in the direction of B, i.e., from a magnetic north to a magnetic south pole, then the middle finger points in the direction of ¥ m . Note here also that on a compass needle, a mark is applied at the magnetic north pole that points to the geographic north pole (a magnetic south pole). Thus, we call a magnet north pole also a north-seeking pole. quantities a and b of Eqs. (1.15a) and (1.15b) as tnv x _ p x _ (l + 5 p ) t a n a _ (l + 5 p ) s i n a a = mv 0 p 0 Vl + tan 2 a + tan 2 β Vi + tan 2 β cos 2 a ' b = mv Σ — Li — . (1 + S P ) tanjö (1 + S P ) sin )ß mv 0 p 0 Vl + tan 2 a 4-tan 2 β Vi 4-tan 2 a cos 2 β ' (2.7a) (2.7b) Here, ρ = p 0 (l + δ ρ ) = V/? 2 + p + p is the momentum of the particle under consideration, and p 0 is the momentum of a reference particle at the same potential. Furthermore, one should make certain that a, b, p x , p 0 , a, and β are all taken at the same position. Note here that in the case of a planar motion β = 0 (Fig. 2.2) or a = 0, the Eqs. (2.7a) and (2.7b) simplify to a = sin a, b = 0, or (2.8a) a = 0, b = sin β. (2.8b) Fig. 2.2. For the case β = 0, the com-ponents of the momentum ρ = ip x + jp z are shown for a particle of mass m and velocity ν = iv x +jv z . 2.2 F o r c e s o n C h a r g e d P a r t i c l e s 31 v Fig. 2.3. The velocities ν and ν + dy o f a particle are s h o w n at t w o consecutive instants. I n a pure magnetic field, d is always perpen-dicular to v.
eBook - PDF- Roger G. Newton(Author)
- 2021(Publication Date)
- Princeton University Press(Publisher)
C H A P T E R 4 Fields and Particles ONE OF MICHAEL FARADAY'S greatest contributions to physics was the introduction of the notion of the field. To appreciate the scope of this concept, recall the discussion in chapter 2 of Newton's sem-inal approach to the problem of how particle A in position P A in-fluences the motion of particle B in position P B ; he broke it into two parts: (1) what force does particle A exert on particle B? and (2) what influence does this force have on the motion of parti-cle B? The answer to the first of these questions introduced the repugnant idea of action at a distance—a force exerted by A at P A on B at the distant point P B . For electric and magnetic influences, Faraday, in effect, subdivided the first question into two further parts: (la) what field does A produce everywhere in space? and (lb) what force does the field at P B exert on particle B at the very point P B ? By doing so, he circumvented altogether the need for a force acting at a distance. This remarkable idea was then general-ized and has pervaded all of fundamental physics ever since. Faraday, who had little interest in mathematics and did not in-vent the field concept for mathematical purposes, imagined electric and magnetic fields in very concrete and physical terms as lines that stretched through space, exerting forces like rubber bands. Maxwell subsequently replaced these images by intricate mechan-ical models of interlocking wheels making up the luminiferous ether that carried light and other electromagnetic waves, but from the time Hertz opened his treatise Electric Waves by declaring that Maxwell's theory is Maxwell's system of equations, we have given up all such pictorial representations and think of fields sim-ply as abstract conditions of space. The notion of an all-pervading ether finally fell victim to Einstein's theory of relativity.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
